Before introducing the Berry phase, we review the elegant mathematical framework behind it.It helps explaining why the Berry phase is often also called the geometric phase.
1.1 Parallel transport and anholonomy angle
Consider a two-dimensional curved surface embedded in a three dimensional Euclidean space.At each pointx = (x1, x2) on the surface, there is a vector spaceTx formed by the tangentvectors at that point. For an ant living on the surface, is it possible to judge if two vectors atdifferent locations (1 and 2) of the surface are nearly parallel or far from it?One possible way to calibrate the difference between two vectors at different locations is asfollows: Starting from point1, the ant can carry the vector around in such a way that it makesa fixed relative angle with the tangent vector along a path between 1 and 2 (see Fig. 1a). Sucha vector is said to beparallel transported. One can then compare the vector already at point 2with the parallel transported vector for difference.Notice that, if we follow this rule, then “being parallel” isa path-dependent concept. That is,one cannot have a global definition of “being parallel” on thecurved surface. The other way tosay the same thing is that, if you parallel transport a vectoralong a closed loop on the surface,then the final vectorvf is generically different from the initial vectorvi (see Fig. 1b).The angle between these two vectors is called the anholonomyangle (or defect angle). Suchan angle is an indication of how curved the surface is. One canuse it to define the intrinsiccurvature of the surface. For example, for a sphere with radiusR, the defect angleα for avector transported around a spherical triangle is equal to the solid angleΩ subtended by thistriangle,
α = Ω =A
R2, (1)
whereA is the area enclosed by the triangle.One can define the curvature at pointx as the ratio betweenα andA for an infinitesimallyclosed loop aroundx. According to this definition, the sphere has a constant curvature1/R2
everywhere on the surface.You can apply the same definition to find out the intrinsic curvature of a cylinder. The resultwould be zero. That is, the cylinder has no intrinsic curvature. That is why we can cut it openand lay it down on top of a desk easily without stretching.
1.2 Moving frame and curvature
In practice, apart from a few simple curved surfaces, it is not easy to determine the curvaturewithout using algebraic tools. At this point, it helps introducing the method of the movingframe. We follow a very nice article by M. Berry (see Berry’s introductory article in Ref. [1])and apply this method to calculate the curvature.Instead of moving a vector, one now moves an orthonormal frame (a triad) along a pathCbetween two points. At the starting point, the triad is(r, e1, e2), wherer is the unit vector alongthe normal direction and(e1, e2) is an orthonormal basis of the tangent vector spaceTx.As a rule of parallel transport, we require that, when movingalongC, the triad should not twistaroundr. That is, ifω is the angular velocity of the triad, then
ω · r = 0. (2)
X5.3
α12
34
1 2
(a) (b)
v1 v2
vi
vf
Fig. 1: (a) Parallel transport of a vector from 1 to 2. It offers a way to comparev1 andv2 on acurved surface. (b) A vector is parallel transported arounda closed path. When the surface iscurved, the final vector would point to a different directionfrom the initial vector. The angle ofdifferenceα is called the anholonomy angle.
Using the identity˙e1 = ω × e1 it follows from this requirement thate1 · e2 = 0:
ω · r = ω · e1 × e2
= ω × e1 · e2 = ˙e1 · e2 = 0. (3)
Likewise also the relatione2 · e1 = 0 is shown easily.To make further analogy with the complex quantum phase in thenext section, let us introducethe following complex vector,
ψ =1√2
(e1 + ie2) . (4)
Then the parallel transport condition can be rephrased as,
Im(
ψ∗ · ψ)
= 0, or iψ∗ · ψ = 0. (5)
Notice that the real part ofψ∗ · ψ is always zero sincee1 · e1 ande2 · e2 are time independent.Instead of the moving triad, we could also erect a fixed triad,(r, u, v), at each point of thesurface and introduce
n =1√2
(u+ iv) . (6)
Assuming these two triads differ by an angleα(x) (around ther-axis), thenψ(x) = n(x)e−iα(x).It follows that
ψ∗ · dψ = n∗ · dn− idα. (7)
Because of the parallel transport condition in Eq. (5), one hasdα = −in∗ ·dn. Finally, the twistangle accumulated by the moving triad after completing a closed loopC is,
α(C) = −i∮
C
n∗ · dndxdx, (8)
where we have changed the variable of integration to the coordinate on the surface. Therefore,the defect angle can be calculated conveniently using the fixed-triad basis.
X5.4 Ming-Che Chang
With the help of the Stokes theorem, one can transform the line integral to a surface integral,
α(C) =
∫
S
1
i
(
dn∗
dx1
· dndx2
− dn∗
dx2
· dndx1
)
dx1dx2, (9)
whereS is the area enclosed byC. In the case of the sphere, one can choose(x1, x2) to bethe spherical coordinates(θ, φ), and chooseu andv to be the unit vectorsθ andφ in sphericalcoordinates. That is,u = (cos θ cosφ, cos θ sin φ,− sin θ) andv = (− sin φ, cosφ, 0). It is notdifficult to show that the integrand in Eq. (9) issin θdθdφ. Therefore,α(C) is indeed the solidangle of the areaS.The integral in Eq. (9) over the whole sphere (the total curvature) is equal to its solid angle,4π. In fact, any closed surface that has the same topology as a sphere would have the sametotal curvature2π × 2. The value of 2 (Euler characteristic) can thus be regarded as a numbercharacterizing the topology of sphere-like surfaces. In general, for a closed surface withg holes,the Euler characteristic is2 − 2g. For example, the total curvature of a donut (g = 1) is 0. Thisis the beautiful Gauss-Bonnet theorem in differential geometry.
2 Anholonomy in quantum mechanics
Similar to the parallel transported vector on a curved surface, the phase of a quantum state (notincluding the dynamical phase) may not return to its original value after a cyclic evolution inparameter space. This fact was first exposed clearly by Michael Berry [3] in his 1984 paper. Inthis section, we introduce the basic concept of the Berry phase, in later sections we will moveon to examples of the Berry phase in condensed matter.
2.1 Introducing the Berry phase
Let us start from a time-independentsystem described by a HamiltonianH(r,p). We denote theeigenstates by|m〉 and the eigenvalues byǫm. For simplicity, the energy levels are assumed to benon-degenerate. An initial state|ψ0〉 =
∑
am|m〉 evolves to a state|ψt〉 =∑
ame−i/~ǫmt|m〉
at timet. The probability of finding a particle in a particular level remains unchanged, eventhough each level acquires a different dynamical phasee−i/~ǫmt. In particular, if one starts withan eigenstate of the Hamiltonian,|ψ0〉 = |n〉, with am = δm,n, then the probability amplitudedoes not “leak” to other states.Let us now consider a slightly more complicated system with two sets of dynamical variablesH(r,p;R,P). The characteristic time scale of the upper-case set is assumed to be much longerthan that of the lower-case set. For example, the system can be a diatomic moleculeH+
2 . Theelectron and nuclei positions are represented byr andR respectively. Because of its largermass, the nuclei move more slowly (roughly by a thousand times) compared to the electron. Inthe spirit of the Born-Oppenheimer approximation, one can first treatR as a time-dependentparameter, instead of a dynamical variable, and study the system at each “snapshot” of theevolution. The kinetic part of the slow variable is ignored for now.Since the characteristic frequency of the nuclei is much smaller than the electron frequency, anelectron initially in an electronic state|n〉 remains essentially in that state after timet,
|ψt〉 = eiγn(R)e−i/~R t0
dtǫn(Rt)|n;R〉. (10)
X5.5
Table 1: Anholonomies in geometry and quantum state
anholonomy anholonomy angle Berry phaseclassification of topology Euler characteristic Chern number
Apart from the dynamical phase, one is allowed to add an extraphaseeiγn(R) for each snapshotstate. Such a phase is usually removable by readjusting the phase of the basis|n;R〉 [2]. In1984, almost six decades after the birth of quantum mechanics, Berry [3] pointed out that thisphase, like the vector in the previous section, may not return to its original value after a cyclicevolution. Therefore, it is not always removable.To determine this phase, one substitutes Eq. (10) into the time-dependentSchrodinger equation.It is not difficult to get an equation forγn(t),
γn(t) = i〈n|n〉. (11)
Therefore, after a cyclic evolution, one has
γn(C) = i
∮
C
〈n| ∂n∂R
〉 · dR =
∮
C
A · dR, (12)
whereC is a closed path in theR-space. The integrandA(R) ≡ i〈n| ∂n∂R
〉 is often called theBerry connection.If the parameter space is two dimensional, then one can use Stokes’ theorem to transform theline integral to a surface integral,
γn(C) = i
∫
S
〈 ∂n∂R
| × | ∂n∂R
〉 · d2R =
∫
S
F · d2R. (13)
The integrandF(R) ≡ ∇R × A(R) is usually called theBerry curvature. For parameterspaces with higher dimensions, such a transformation can still be done using the language ofthe differential form.By now, the analogy between Eqs. (8,9) and Eqs. (12,13) should be clear. Notice that|n〉 is anormalized basis with〈n|n〉 = 1. Therefore,〈n|n〉 should be purely imaginary andi〈n|n〉 is areal number. The basis state|n〉 plays the role of the fixed triadn in the previous subsection.Both are single-valued. On the other hand, the parallel transported state|ψ〉 and the movingtriadψ are not single-valued.A point-by-point re-assignment of the phase of the basis state, |n;R〉′ = eig(R)|n;R〉, changesthe Berry connection,
A′ = A− ∂g
∂R. (14)
However, the Berry curvatureF and the Berry phase are not changed. This is similar to thegauge transformation in electromagnetism: one can choose different gauges for the potentials,but the fields are not changed. Such an analogy will be explored further in the next subsection.
X5.6 Ming-Che Chang
θ
φ
S
x
y
z
Fig. 2: A long solenoid hinged at the origin is slowly rotating around thez-axis. At each instant,the spin at the origin aligns with the uniform magnetic field inside the solenoid.
A short note: It is possible to rephrase the anholonomy of thequantum state using the mathemat-ical theory of fiber bundles, which deals with geometrical spaces that can locally be decomposedinto a product space (the “fiber” space times the “base” space), but globally show nontrivialtopology. The Mobius band is the simplest example of such a geometric object: Locally it is aproduct of two one-dimensional spaces but globally it is not(because of the twisting). In ourcase, the fiber is the space of the quantum phaseγ(R) and the base is the space ofR. The con-cept of the parallel transport, the connection, and the curvature all can be rephrased rigorouslyin the language of fiber bundles [4]. Furthermore, there is also a topological number (similar tothe Euler characteristic) for the fiber bundle, which is called the Chern number.The analogy between geometric anholonomy and quantum anholonomy is summarized in Ta-ble 1.
2.2 A rotating solenoid
To illustrate the concept of the Berry phase, we study a simple system with both slow and fastdegrees of freedom. Following M. Stone [5], we consider a rotating (long) solenoid with anelectron spin at its center. The solenoid is tilted with a fixed angleθ and is slowly gyratingaround thez-axis (see Fig. 2). Therefore, the electron spin feels a uniform magnetic field thatchanges direction gradually. This example is a slight generalization of the spin-in-magnetic-field example given by Berry in his 1984 paper. The Hamiltonian of this spin-in-solenoid systemis,
H =L2
2I+ µBσ · B, (15)
whereL andI are the angular momentum and the moment of inertia of the solenoid, respec-tively, and the Bohr magneton isµB = e~/2mc.The magnetic fieldB along the direction of the solenoid is our time-dependent parameterR. Inthe quasi-static limit, the rotation energy of the solenoidis neglected. When the solenoid rotatesto the angle(θ, φ), the spin eigenstates are
|+; B〉 =
(
cos θ2
eiφ sin θ2
)
, |−; B〉 =
(
−e−iφ sin θ2
cos θ2
)
. (16)
X5.7
Table 2: Analogy between electromagnetism and quantum anholonomy
magnetic monopole point degeneracymagnetic fluxΦ(C) Berry phaseγ(C)
These states can be obtained, for example, from the spin-up (-down) states|±〉 by a rotatione−iσ·θ(θ/2), in which the rotation axisθ = (− sinφ, cosφ, 0) is perpendicular to bothz andB.Using the definitions of the Berry connection and the Berry curvature in Eqs. (12) and (13), oneobtains
A± = ∓1
2
1 − cos θ
B sin θφ (17)
F± = ∓1
2
B
B2. (18)
They have the same mathematical structure as the vector potential and the magnetic field of amagnetic monopole. The location of the “monopole” is at the origin of the parameter space,where a point degeneracy occurs. The strength of the monopole (1/2) equals the value of thespin (this is true for larger spins also). That is why the Berry connection and the Berry curvatureare sometimes called the Berry potential and the Berry field.In this picture, the Berry phase isequal to the flux of the Berry field passing through a loopC in parameter space. It is easy tosee that,
γ±(C) = ∓1
2Ω(C), (19)
whereΩ(C) is the solid angle subtended by loopC with respect to the origin. The similaritybetween the theory of Berry phase and electromagnetism is summarized in Table 2.The Berry phase of the fast motion is only half of the story. When the quantum state of the fastvariable acquires a Berry phase, there will be an interesting “back action” to the slow motion.For example, for the rotating solenoid, the wave function ofthe whole system can be expandedas
|Ψ〉 =∑
n=±
ψn(R)|n;R〉, (20)
in which ψn(R) describes the slow quantum state. From the Schrodinger equation,H|Ψ〉 =E|Ψ〉, one can show that,
[
~2
2I sin2 θ
(
1
i
d
dφ− An
)2
+ ǫn
]
ψn = Eψn, (21)
whereǫn is the eigen-energy for the fast degree of freedom, andAn ≡ i〈n;R| ddφ|n;R〉. The
off-diagonal coupling between|+〉 and|−〉 has been ignored. Therefore, the effective Hamil-tonian for the slow variable acquires a Berry potentialAn(R). Such a potential could shiftthe spectrum and results in a force (proportional to the Berry curvature) upon the slow motion,much like the effect of vector potentialA(r) and magnetic field on a charged particle.
X5.8 Ming-Che Chang
(a) (b)
B B
S S
e-
Fig. 3: (a) A metal ring in a non-uniform magnetic field. The spin of the electron that is circlingthe ring would align with the magnetic field and trace out a solid angle in its own referenceframe. (b) A ferromagnetic ring in a non-uniform magnetic field. The spins on the ring are bentoutward because of the magnetic field.
3 Berry phase and spin systems
A natural place to find the Berry phase is in spin systems. Numerous researches related to thissubject can be found in the literature [6]. Here we only mention two examples, one is relatedto the persistent spin current in a mesoscopic ring, the other relates to quantum tunneling in amagnetic cluster.
3.1 Persistent spin current
We know that an electron moving in a periodic system feels no resistance. The electric resis-tance is a result of incoherent scatterings from impuritiesand phonons. If one fabricates a cleanone-dimensional wire, wraps it around to form a ring, and lowers the temperature to reduce thephonon scattering, then the electron inside feels like living in a periodic lattice without electricresistance.For such a design to work, two ingredients are essential: First, the electron has to remain phasecoherent (at least partially) after one revolution. Therefore, a mesoscopic ring at very low tem-perature is usually required. Second, to have a traveling wave, there has to be a phase advance(or lag) after one revolution. This can be achieved by threading a magnetic fluxφ through thering, so that the electron acquires an Aharonov-Bohm (AB) phase(e/~)φ = 2π(φ/φ0) afterone cycle, whereφ0 is the flux quantumh/e. When this does happen, it is possible to observethe resultingpersistent charge currentin the mesoscopic ring.Soon after this fascinating phenomenon was observed [7], itwas proposed that, in addition tothe AB phase, a spinful electron can (with proper design) acquire a Berry phase after one cycle,and this can result in a persistentspincurrent [8]. The design is as follows: Instead of a uniformmagnetic field, a textured magnetic field is used, so that during one revolution, the electron spinfollows the direction of the field and traces out a non-zero solid angleΩ (see Fig. 3a). Accordingto Eq. (19), this gives rise to a spin-dependent Berry phaseγσ(C) = −(σ/2)Ω, whereσ = ±.After combining this with the (spin-independent) AB phase,spin-up and spin-down electronshave different phase shifts, generating different amountsof persistent particle currentI+, I−.Therefore, a spin current defined asIs = (~/2)(I+ − I−) is not zero.
X5.9
IS
Ω/4π
1/2-1/2
T=0
T>0
Fig. 4: Persistent spin current as a function of the solid angle. At non-zero temperature, thesharp edges of the sawtooth become smooth.
To illustrate the physics just mentioned, consider a ring that allows only angular motion. Beforeapplying the magnetic flux, the electron with wave vectork picks up a phasekL from circlingthe ring, whereL = 2πR andR is the radius of the ring. Because of the periodic boundarycondition, one haskL = 2πn (n ∈ Z). After adding the AB phase and the Berry phase, itbecomeskL = 2πn + 2π(φ/φ0) − σ(Ω/2). Therefore, the energy of an electron in then-thmode is
ǫnσ =~
2k2
2m+ µBBσ =
~2
2mR2
(
n+φ
φ0
− σφΩ
φ0
)2
+ µBBσ, (22)
whereφΩ/φ0 ≡ Ω/4π.The spin current can be calculated from
Is =1
L
∑
n,σ
(
~
2σ
)
∂ǫnσ
~∂kPnσ, (23)
wherePnσ = exp(−ǫnσ/kBT )/Z is the probability of the electron in the(n, σ)-state, andZ =∑
n,σ e−ǫnσ/kBT . For a particulark andφ, the current can also be written as
Is = −∑
n,σ
∂ǫnσ
∂ΩPnσ. (24)
To get a rough understanding, we consider the simplest case,where then = 1 mode is populatedwith equal numbers of spin-up and -down electrons (if the Zeeman splitting is negligible). Thehigher modes are all empty at low enough temperature. In thiscase, the spin currentIs =−(~2/4πmR2)(Ω/4π) is proportional to the solid angle of the textured magnetic field (seeFig. 4). At higher temperature, the sawtooth curve will become smooth.The mesoscopic ring considered above is a metal ring with moving electrons that carry the spinswith them. A different type of spin current has also been proposed in a ferromagnetic ring withno moving charges [9]. Again the ring is subject to a texturedmagnetic field, such that whenone moves round the ring, one sees a changing spin vector thattraces out a solid angleΩ (seeFig. 3b). As a result, the spin wave picks up a Berry phase whentraveling around the ring,resulting in a persistent spin current. So far neither type of persistent spin current has beenobserved experimentally.
X5.10 Ming-Che Chang
3.2 Magnetic cluster
Berry phase plays a dramatic role in the quantum tunneling ofnano-sized magnetic clusters.The tunneling between two degenerate spin states of the cluster depends on whether the totalspin of the particle is an integer or a half-integer. In the latter case, the tunneling is completelysuppressed because different tunneling paths interfere destructively as a result of the Berryphase [10].Consider a single-domain ferromagnetic particle without itinerant spin. Its total spinJ can beof order ten or larger, as long as tunneling is still possible. Assume that the particle lives in ananisotropic environment with the Hamiltonian,
H = −k1J2
z
J2+ k2
(
J2x
J2−J2
y
J2
)
, (k1 > k2). (25)
That is, the easy axis is along thez-axis and the easy plane is theyz-plane. The cluster is inthe ground state when the spin points to the north pole or to the south pole of the Bloch sphere.Even though these two degenerate states are separated by a barrier, the particle can switch itsdirection of spin via quantum tunneling.To study the Berry phase effect on the tunneling probability, the best tool is the method of pathintegrals. In the following, we give a brief sketch of its formulation.The fully polarized spin state|n, J〉 along a directionn with spherical angles(θ, φ) can bewritten as,
|n, J〉 = |n,+〉 ⊗ |n,+〉 · · · ⊗ |n,+〉
=2J∏
l=1
e−i θ2σl·θ|z,+〉l, (26)
where|n,+〉 is the spin-1/2 “up” state along then-axis andθ is a unit vector along thez× n di-rection. Such a so-calledspin coherent statecan be used to “resolve” the identity operator [11],
I =2J + 1
4π
∫
dΩ|n〉〈n|, (27)
where|n〉 is an abbreviation of|n, J〉.In order to calculate the transition probability amplitude〈nf | exp[−(i/~)HT ]|ni〉, one first di-vides the time evolution into steps,exp(−i/~HT ) = [exp(−i/~Hdt)]N , dt = T/N , then insertthe resolution of identity in Eq. (27) between neighboring steps. The transition amplitude thenbecomes a product of factors with the following form,
〈n(t+ dt)|e− i~Hdt|n(t)〉 ≃ 〈n(t+ dt)|n(t)〉 − i
~〈n(t+ dt)|H(J)|n(t)〉dt
≃ 1 − 〈n| ˙n〉dt− i
~H(Jn)dt. (28)
In the final step, we have replaced the quantum Hamiltonian bya classical Hamiltonian. Thatis, 〈H(J)〉 = H(〈J〉). This holds exactly if the Hamiltonian is linear inJ, but is only anapproximation in general. The correction due to the non-commutativity of the spin operator isroughly of the fraction1/J and can be ignored for large spins.
X5.11
x
y
z
B
easy
hard
medium
Fig. 5: According to the Hamiltonian in Eq. (25), thez-axis and thex-axis are the easy axis andthe hard axis, respectively. There are two (degenerate) ground states at the north pole and thesouth pole of the Bloch sphere. Tunneling from one ground state to the other follows the dashedline on they − z plane. Applying a magnetic field along thex-direction moves the locations ofthe ground states and shrinks the tunneling path to a smallerloop.
Finally, by summing over paths in then-space, one has
〈nf |e−i~HT |ni〉 =
∫
[Dn] exp
i
~
∫ tf
ti
[
i~〈n| ˙n〉 −H(Jn)]
dt
. (29)
Notice that the first integral in the exponent generates a Berry phase for a path (see Eq. (12)). Inthe semiclassical regime, the functional integral in Eq. (29) is dominated by the classical pathnc with least action, which is determined from the dynamical equation ofn (see below). Duringtunneling, the paths under the barrier are classically inaccessible andn becomes an imaginaryvector. It is customary to sacrifice the reality of timet to keepn real. The good news is that thefinal result does not depend on which imaginary world you choose to live in.Defineτ = it, then the transition amplitude dominated by the classical action is,
〈nf |e−i~
HT |ni〉 ∝ eiR f
iA·dnce−1/~
R f
iH(Jnc)dτ , (30)
whereA = i〈n|∇n〉 is the Berry potential. The integral of the Berry potential is gauge depen-dent if the path is open. It is well defined for a closed loop, such as the classical path on theyz-plane in Fig. 5. The Berry phase for such a loop is2πJ since it encloses an area with solidangle2π (Cf. Eq. (19)). This is also the phase difference between thetwo classical paths fromthe north pole to the south pole. Therefore,
〈−z|e− i~HT |z〉 ∝ cos(πJ)e−1/~
R f
iH(Jnc)dτ . (31)
WhenJ is a half integer, the transition process is completely suppressed because of the Berryphase. The conclusion remains valid if one considers classical paths with higher winding num-bers [10].As a reference, we also write down the equation of motion fornc that is determined from theclassical action in Eq. (30),
Jdn
dt= n× ∂H(Jn)
∂n. (32)
X5.12 Ming-Che Chang
P
P
ΔP
Unit cell
+ (a)
(b)
(c)
+Fig. 6: An one-dimensional solid with infinite length. Different choices of the unit cell givedifferent electric polarization vectors ((a), (b)). On theother hand, the change of polarizationdoes not depend on the choice of the unit cell (c).
This is the Bloch equation for spin precession, in which∂H/∂n plays the role of an effectivemagnetic field.One comment is in order: One can apply a magnetic field along thex-axis that shifts the energyminima along that direction and shrinks the classical loop (see Fig. 5). In an increasinglystronger field, the size of the loopC eventually would shrink to zero. That is, the Berry phaseγC would decrease from the maximum value of2πJ to zero. During the process, one expectsto encounter the no-tunneling situation several times wheneverγC/2π hits a half-integer. Sucha dramatic Berry phase effect has been observed [12].
4 Berry phase and Bloch state
In the second half of this article, we focus on the Berry phasein periodicsolids. It has been play-ing an ever more important role in recent years due to severaldiscoveries and “re-discoveries”,in which the Berry phase either plays a crucial role or offersa fresh perspective.
4.1 Electric polarization
It may come as a surprise to some people that the electric polarizationP of an infinite periodicsolid (or a solid with periodic boundary conditions) is generically not well defined. The reasonis that, in a periodic solid, the electric polarization depends on your choice of the unit cell(see Fig. 6a,b). The theory of electric polarization in conventional textbooks applies only tosolids consisting of well localized charges, such as ionic or molecular solids (Clausius-Mossottitheory). It fails, for example, in a covalent solid with bondcharges such that no natural unit cellcan be defined.A crucial observation made by R. Resta [13] is that, even though the value ofP may be am-biguous, its change is well defined (see Fig. 6c). It was laterpointed out by King-Smith andVanderbilt [14] that∆P has a deep connection with the Berry phase of the electronic states.The outline of their theory below is based on one-particle states. However, the same scheme
X5.13
applies to real solids with electronic interactions, as long as one replaces the one-particle statesby the Kohn-Sham orbitals in the density functional theory.We will useλ to label the degree of ion displacement. It varies from 0 to 1 as the ions shiftadiabatically from an initial state to a final state. The difference of polarizations between thesetwo states is given by
∫ 1
0dλdP/dλ, where
P(λ) =q
V
∑
i
〈φi|r|φi〉. (33)
The summation runs over filled Bloch statesφi (with λ-dependence) andV is the volume ofthe material. For an infinite crystal, the expectation valueof r is ill-defined. Therefore, weconsider a finite system at first, and letV → ∞ when the mathematical expression becomeswell-defined.The Bloch states are solutions of the Schrodinger equation,
Hλ|φi〉 =
(
p2
2m+ Vλ
)
|φi〉 = ǫi|φi〉, (34)
whereVλ is the crystal potential. From Eq. (34), it is not difficult toshow that, forj 6= i, onehas
(ǫi − ǫj) 〈φj|∂φi
∂λ〉 = 〈φj|
∂Vλ
∂λ|φi〉. (35)
Therefore,dP
dλ=
q
V
∑
i
∑
j 6=i
[
〈φi|r|φj〉〈φj|V ′
λ|φi〉ǫi − ǫj
+H.c.
]
. (36)
There is a standard procedure to convert the matrix elementsof r to those ofp: Start with thecommutation relation,[r, Hλ] = i~p/m, and sandwich it between thei-state and thej-state(againj 6= i), we can get an useful identity,
〈φi|r|φj〉 =i~
m
〈φi|p|φj〉ǫj − ǫi
. (37)
With the help of this identity, Eq. (36) becomes the following expression derived by Resta [13],
dP
dλ=
q~
imV
∑
i
∑
j 6=i
[〈φi|p|φj〉〈φj|V ′λ|φi〉
(ǫi − ǫj)2−H.c.
]
. (38)
Now all of the matrix elements are well-defined and the volumeV can be made infinite. Afterintegrating with respect toλ, the resulting∆P is free of ambiguity, even for an infinite covalentsolid.For Bloch states, the subscripts arei = (m,k) andj = (n,k), wherem,n are the band indicesandk is the Bloch momentum defined in the first Brillouin zone. Eq. (38) can be transformed toa very elegant form, revealing its connection with the Berrycurvature [14]. One first defines ak-dependent Hamiltonian,H = e−ik·rHeik·r. It is the Hamiltonian of the cell-periodic functionunk. That is,H|unk〉 = ǫnk|unk〉, whereφnk = eik·runk. It is then straightforward to show that,
〈φmk|p|φnk〉 =m
~〈umk|
[
∂
∂k, H
]
|unk〉 =m
~(ǫnk − ǫmk)〈umk|
∂unk
∂k〉. (39)
X5.14 Ming-Che Chang
With the help of this equation and another one very similar toEq. (35) (just replace theφi’s bytheui’s), we finally get (α = x, y, z)
dPα
dλ= −iq
V
∑
nk
(⟨
∂unk
∂kα|∂unk
∂λ
⟩
−⟨
∂unk
∂λ|∂unk
∂kα
⟩)
= − q
V
∑
nk
Ωnkαλ(k), (40)
whereΩnkαλ ≡ i
(
〈 ∂u∂kα
|∂u∂λ〉 − c.c.
)
is the Berry curvature for then-th band in the parameter
space ofkα andλ (Cf. Eq. (13)).Let us take a one-dimensional system as an example. Assumingthe lattice constant isa. Thenthe difference of polarization is (q = −e),
∆P =e
2π
∑
n
∫ 2π/a
0
dk
∫ 1
0
dλΩnkλ. (41)
The area of integration is a rectangle with lengths1 and2π/a on each side. The area integralcan be converted to a line integral around the boundary of therectangle, which gives the Berryphaseγn of such a loop. Therefore,
∆P = e∑
n
γn
2π. (42)
In the special case where the final state of the deformationV1 is the same as the initial stateV0,the Berry phaseγn can only be integer multiples of2π [14]. Therefore, the polarizationP for acrystal state is uncertain by an integer chargeQ.One the other hand, this integer chargeQ does carry a physical meaning when it is the difference∆P between two controlled states. For example, when the lattice potential is shifted by onelattice constant to the right, thisQ is equivalent to the total charge being transported. Basedon such a principle, it is possible to design a quantum chargepump using a time-dependentpotential [15].
4.2 Quantum Hall effect
The quantum Hall effect (QHE) has been discovered by K. von Klitzing et al. [16] in a two-dimensional electron gas (2DEG) at low temperature and strong magnetic field. Under suchconditions, the Hall conductivityσH develops plateaus in theσH(B) plot. For the integer QHE,these plateaus always locate at integer multiples ofe2/h to great precision, irrespective of thesamples being used. Such a behavior is reminiscent of macroscopic quantum phenomena, suchas the flux quantization in a superconductor ring.To explain the integer QHE, Laughlin wraps the sheet of the 2DEG to a cylinder to simulatethe superconductor ring, and studies the response of the current with respect to a (fictitious)magnetic flux through the cylinder (see Fig. 7). He found that, as the flux increases by oneflux quantumh/e, integer chargesQ = ne are transported from one edge of the cylinder tothe other [17]. This charge transport in the transverse direction gives the Hall current, and theintegern can be identified with the integer of the Hall conductancene2/h [18].
X5.15
Ix
Vy
B
B
B
B
ψVy
Fig. 7: In Laughlin’s argument, the 2DEG is on the surface of a cylinder. The real magneticfield B now points radially outward. In addition, there is a fictitious flux threading through thecylinder. When the fictitious flux changes by one flux quantum,integer number of electrons arebe transported from one edge of the cylinder to the other.
Soon afterwards, Thoulesset al. (TKNdN) [19] found that the Hall conductivity is closelyrelated to the Berry curvature (not yet discovered by Berry at that time) of the Bloch state. Wenow briefly review the TKNdN theory.Consider a 2DEG subject to a perpendicular magnetic field anda weak in-plane electric field.In order not to break the periodicity of the scalar potential, we choose a time-dependent gaugefor the electric field. That is,E = −∂AE/∂t, AE = −Et. The Hamiltonian is,
H =(π − eEt)2
2m+ VL(r), (43)
whereπ = p+ eA0 has included the vector potential of the magnetic field, andVL is the latticepotential. Similar to the formulation the in previous subsection, it is convenient to use thek-dependent HamiltonianH and the cell-periodic functionunk in our discussion. They are relatedby H|unk〉 = Enk|unk〉.We will assume that the system can be solved with known eigenvalues and eigenstates,H0|u(0)
nk〉 =
E(0)nk |u
(0)nk〉 in the absence of an external electric field [20]. The electric field is then treated as a
perturbation. To the first-order perturbation, one has
The velocity of a particle in then-th band is given byvn(k) = 〈unk|∂H/~∂k|unk〉. Aftersubstituting the states in Eq. (44), we find
vn(k) =∂ǫn~∂k
− i∑
n′ 6=n
(
〈n|∂H∂k
|n′〉〈n′|∂n∂t〉
ǫn − ǫn′
− c.c.
)
. (45)
The first term is the group velocity in the absence of the electric perturbation. With the help ofan equation similar to Eq. (39),
〈n|∂H∂k
|n′〉 = (ǫn − ǫn′) 〈∂n∂k
|n′〉, (46)
X5.16 Ming-Che Chang
one finally gets a neat expression,
vn(k) =∂ǫn~∂k
− i
(⟨
∂n
∂k|∂n∂t
⟩
−⟨
∂n
∂t|∂n∂k
⟩)
. (47)
By a change of variable, the second term becomesΩn × k = −(e/~)Ωn × E, whereΩnα =iǫαβγ〈 ∂n
∂kβ| ∂n∂kγ
〉 is the Berry curvature in momentum space.
For a 2DEG,Ωn = Ωnz. All states below the Fermi energy contribute to the currentdensity,
j =1
V
∑
nk
−evn(k) =e2
~
∑
n
∫
d2k
(2π)2Ωn(k) × E. (48)
Notice that the first term in Eq. (47) does not contribute to the current. From Eq. (48), it is clearthat the Hall conductivity is given by,
σyx =e2
h
∑
n
1
2π
∫
d2kΩn(k). (49)
Thoulesset al. have shown that the integral of the Berry curvature over the whole BZ di-vided by2π must be an integercn. Such an integer (the Chern number mentioned in Sec. 2.2)characterizes the topological property of the fiber bundle space, in which the base space is thetwo-dimensional BZ, and the fiber is the phase of the Bloch state (see the discussion near theend of Sec. 2.1). Therefore, the Hall conductivity of a filledband is always an integer multipleof e2/h. Such a topological property is the reason why the QHE is so robust against disordersand sample varieties. Even though the discussion here is based on single-particle Bloch states,the conclusion remains valid for many-body states [21].Some comments are in order. First, the formulas behind the change of electric polarization∆P
in Sec. 4.1 and those of the quantum Hall conductivity here look very similar. Both are basedon the linear response theory. In fact, the analogy can be carried further if∆P is considered asthe time integral of a polarization currentjP = ∂P/∂t. The latter, similar to the quantum Hallcurrent in Eq. (48), can be related to the Berry curvature directly.Second, if a solid is invariant under space inversion, then the cell-periodic state has the symme-try,
un−k(−r) = unk(r). (50)
On the other hand, if the system has time-reversal symmetry,then
u∗n−k(r) = unk(r). (51)
As a result, if both symmetries exist, then one can show that the Berry potentialAn = i〈n|∂n∂k〉
(and therefore the Berry curvature) is zero for allk. The conclusion, however, does not hold ifthere is band crossing or spin-orbit interaction (not considered so far).That is, the Berry potential (or curvature) can be non-zero if (i) the lattice does not have spaceinversion symmetry. This applies to the polarization discussed in the previous subsection. (ii)Time-reversal symmetry is broken, e.g., by a magnetic field.This applies to the quantum Hallsystem in this subsection. In the next subsection, we consider a system with spin-orbit interac-tion, in which the Berry curvature plays an important role.
X5.17
saturation slope=RN
B
H
0RAHMS
Fig. 8: When one increases the magnetic field, the Hall resistivity of a ferromagnetic materialrises quickly. It levels off after the sample is fully magnetized.
4.3 Anomalous Hall effect
Soon after Edwin Hall discovered the effect that bears his name in 1879 (at that time he wasa graduate student at Johns Hopkins university), he made a similar measurement on iron foiland found a much larger Hall effect. Such a Hall effect in ferromagnetic materials is called theanomalous Hall effect (AHE).The Hall resistivity of the AHE can be divided into two terms with very different physics (pro-posed by Smith and Sears in 1929) [22],
ρH = ρN + ρAH = RN (T )B + RAH(T )µ0M(T,H), (52)
whereB = µ0(H + M). The first (normal) term is proportional to the magnetic fieldin thesample. The second (anomalous) term grows roughly linearlywith the magnetizationM andthe coefficientRAH is larger thanRN by one order of magnitude or more. If the applied fieldis so strong that the material is fully magnetized, then there is no more enhancement from theanomalous term and the Hall coefficient suddenly drops by orders of magnitude (see Fig. 8).Since the normal term is usually much smaller than the anomalous term, we will neglect it inthe following discussion.Unlike the ordinary Hall effect, the Hallresistivityin the AHE increases rapidly with tempera-ture. However, the Hallconductivity,
σH =ρH
ρ2L + ρ2
H
≃ ρH
ρ2L
(if ρL ≫ ρH), (53)
shows less temperature dependence, whereρL is the longitudinal resistivity. The reason willbecome clear later.Since the AHE is observed in ferromagnetic materials, the magnetization (or the majority spin)must play a role here. Also, one needs the spin-orbit (SO) interaction to convert the direction ofthe magnetization to a preferred direction of the transverse electron motion.Among many attempts to explain the AHE, there are two popularexplanations [23], both involvethe SO interaction,
HSO = − ~
4m2c2σ · (p×∇V ). (54)
The first theory was proposed by Karplus and Luttinger (KL) in1954 [24]. It requires noimpurity (the intrinsic scenario) and theV in Eq. (54) is the lattice potential. The Hall resistivity
X5.18 Ming-Che Chang
ρAH is found to be proportional toρ2L. The other explanation is proposed by Smit in 1958 [25].
It requires (non-magnetic) impurities (the extrinsic scenario) andV is the impurity potential. ItpredictsρAH ∝ ρL. When both mechanisms exist, one has
ρAH = a(M)ρL + b(M)ρ2L. (55)
The Smit term is a result of the skewness of the electron-impurity scattering due to the SOinteraction. That is, the spin-up electrons prefer scattering to one side, and the spin-downelectrons to the opposite side. Because of the majority spins of the ferromagnetic state, suchskew-scatterings produce a net transverse current. Smit’sproposal started as an opposition toKL’s theory and gained popularity in the early years. As a result, the KL scenario seems to havebeen ignored for decades.At the turn of this century, however, several theorists picked up the KL theory and put it underthe new light of the Berry curvature [26]. Subsequently, increasing experimental evidencesindicate that, in several ferromagnetic materials, the KL mechanism does play a much moreimportant role than the skew-scattering. These works published in renowned journals haveattracted much attention, partly because of the beauty of the Berry curvature scenario.KL’s theory, in essence, is very similar to the ones in the previous two subsections. One canfirst regard the Hamiltonian with the SO interaction as solvable, then treat the electric field as aperturbation. To the first order of the perturbation, one canget the electron velocity with exactlythe same form as the one in Eq. (47). The difference is that thestate|n〉 now is modified bythe SO interaction and the solid is three dimensional. That is, one simply needs to consider aperiodic solid without impurities and apply the Kubo formula, which (in these cases) can bewritten in Berry curvatures,
σAH =e2
~
1
V
∑
n,k
Ωn(k). (56)
However, not every solid with the SO interaction has the AHE.The transverse velocities (alsocalled the anomalous velocity) in general have opposite signs for opposite spins in the spin-degenerate bands. Therefore, these two Hall currents will get canceled. Again the ferromagneticstate (which spontaneously breaks the time reversal symmetry) is crucial for a net transversecurrent.From Eq. (53), one hasρH ≃ ρAH = σHρ
2L. Also, the anomalous current generated from the
Berry curvature is independent of the relaxation timeτ . This explains why the Hall conductivityin the KL theory is proportional toρ2
L.In dilute magnetic semiconductors, one can show thatA(k) = ξS× k for the conduction bandof the host semiconductor, whereξ is the strength of the SO coupling (more details in Sec. 5.2).Therefore,Ω = ∇ × A = 2ξS. In this case, the coefficientb(M) in Eq. (55) is proportionalto M . In ferromagnetic materials with a more complex band structure, the Berry curvatureshows non-monotonic behavior in magnetization. For one thing, in density-functional-theorycalculations, the Berry curvature can be dramatically enhanced when the Fermi energy is near asmall energy gap [27]. However, spin fluctuations may smear out the erratic behavior and leadto a smooth variation (see Fig. 9) [28].The Berry curvature is an intrinsic property of the electronic states. It appears not only at thequantum level, but also in the semiclassical theory of electron dynamics. In the next section,we will see that the QHE, the AHE, and the spin Hall effect can all be unified in the samesemiclassical theory.
X5.19
(a) (b)
Fig. 9: (a) Calculated anomalous Hall conductivity (the intrinsicpart) versus magnetizationfor Mn5Ge3 using different relaxation times. (b) After averaging overlong-wavelength spinfluctuations, the calculated anomalous Hall conductivity becomes roughly linear inM . Theinitials S.S. refers to skew scattering. The figures are fromRef. [28].
5 Berry phase and wave-packet dynamics
When talking about electron transport in solids, people usetwo different languages: It is eitherparticle scattering, mean free path, cyclotron orbit ..., or localized state, mobility edge, Landaulevel ... etc. In this section, we use the first language and treat the electrons as particles withtrajectories. Besides being intuitive, this approach has the following advantage: The electro-magnetic potentials in the Schrodinger equation are oftenlinear inr and diverge with systemsize. Such a divergence can be avoided if the wave function ofthe electron is localized.
5.1 Wave-packet dynamics
Consider an energy band that is isolated from the other bandsby finite gaps. Also, the energyband is not degenerate with respect to spin or quasi-spin. The energy band with internal (e.g.,spin) degrees of freedom is the subject of the next subsection. When inter-band tunneling canbe neglected, the electron dynamics in this energy band can be described very well using awave-packet formalism.The wave packet can be built by a superposition of Bloch states ψnq in bandn (one bandapproximation),
|W 〉 =
∫
BZ
d3qa(q, t)|ψnq〉. (57)
It is not only localized in position space, but also in momentum space,
〈W |r|W 〉 = rc;
∫
BZ
d3qq|a(q)|2 = qc, (58)
whererc andqc are the centers of mass. The shape of the wave packet is not crucial, as long asthe electromagnetic field applied is nearly uniform throughout the wave packet.Instead of solving the Schrodinger equation, we use the time-dependent variational principle tostudy the dynamics of the wave packet. Recall that in the usual (time-independent) variationalprinciple, one first proposes a sensible wave function with unknown parameters, then minimizes
X5.20 Ming-Che Chang
its energy to determine these parameters. Here, the wave packet is parametrized by its centerof mass(rc(t),qc(t)). Therefore, instead of minimizing the energy, one needs to extremize theactionS[C] =
∫
CdtL, which is afunctionalof the trajectoryC in phase space.
One starts from the following effective Langrangian,
Notice the resemblance between thisS[C] and the the action in the coherent-state path integral(Eq. (29)). The Hamiltonian for a Bloch electron in an electromagnetic field is
H =1
2m(p + eA)2 + VL(r) − eφ(r) ≃ H0 − eφ+
e
2mr × p · B, (60)
in whichH0 = p2/2m + VL andφ andA = 12B × r are treated as perturbations. The fields
are allowed to change slowly in space and time, as long as it isapproximately uniform andquasi-static (adiabatic) from the wave packet’s perspective.To evaluate the Lagrangian approximately, one can Taylor-expand the potentials with respect tothe center of the wave packet and keep only the linear terms. Using this gradient approximation,the wave-packet energy〈W |H|W 〉 is evaluated as [29],
E = E0(qc) − eφ(rc) +e
2mL(qc) · B, (61)
whereE0 is the unperturbed Bloch energy of the band under consideration, andL(kc) =〈W |(r− rc) × p|W 〉 is the self-rotating angular momentum of the wave packet.On the other hand, the first term in Eq. (59) can be written as
i~〈W | ddt|W 〉 = ~〈u|idu
dt〉 + ~qc · rc, (62)
in which |u〉 is the unperturbed cell-periodic function. Therefore, theeffective Lagrangian is
L = ~kc · Rc + (~kc − eAc) · rc −E(rc,kc), (63)
where~kc = ~qc + eAc is the gauge-invariant quasi-momentum,Rc = i〈n| ∂n∂kc
〉 is the Berrypotential, andAc = A(rc).Treating bothrc andkc as generalized coordinates and using the Euler-Lagrange equation, itis not very difficult to get the following (coupled) equations of motion (EOM) for the wavepacket [29],
~kc = −eE − erc × B, (64)
~rc =∂E
∂kc− ~kc ×Ωc, (65)
whereΩc = ∇kc×Rc is the Berry curvature of the band under consideration.
Compared to the usual semiclassical EOM in textbooks, thereare two new quantities in Eqs. (64,65),and both lead to important consequences. The first is the Berry curvatureΩ. It generates theso-called anomalous velocity. In the presence of a perturbing electric field, the anomalous ve-locity is eE × Ω, which is perpendicular to the driving electric field and gives rise to, e.g., theAHE.
X5.21
The second is the spinning angular momentumL in Eq. (61). It is closely related to the orbitalmagnetization of a solid [30]. For a spinful wave packet (Sec. 5.2), thisL modifies the elec-tron spin and is the origin of the anomalousg-factor in solids. In fact, starting from Dirac’srelativistic electron theory (which has no explicit spin inthe Hamiltonian), we have shown that,the wave packet in the positive-energy branch of the Dirac spectrum has an intrinsic spinningangular momentum [31]. That is, it explains why an electron has spin.In the semiclassical theory of electron transport, the current density is given by
j = − e
V
∑
nk
f r, (66)
wheref = f0+δf is the distribution function away from equilibrium. The distribution functionf is determined from the Boltzmann equation,
r · ∂f∂r
+ k · ∂f∂k
= −δfτ, (67)
whereτ is the relaxation time. For a homogeneous system in an electric field, δf ≃ τ e~E · ∂f0
∂k,
and
j ≃ − e
V
∑
nk
(
δf∂En
~∂k+ f0
e
~E ×Ωn
)
. (68)
The usual current (the first term) depends on carrier relaxation timeτ through the change of thedistribution functionδf . On the other hand, the second term gives the Hall current. Clearly, thisΩ is also the one in the Kubo formula of QHE and AHE. (The latter involves spin-degenerateband and belongs more properly to the next subsection.)We emphasize that, just like the Bloch energyE0(k), bothΩ(k) andL(k) are intrinsic to theenergy band (not induced by the applied field). They are the three main pillars of band theory.Unlike the Bloch energy that has been around for a very long time, the other two quantities arerelatively new players, but their importance should increase over time.If there is only a magnetic field, then combining Eq. (64) and Eq. (65) gives
~kc =− e
~
∂E∂kc
× B
1 + e~B · Ω . (69)
It describes a cyclotron orbit moving on a plane perpendicular to the magnetic field. The orbitis an energy contour on the Fermi surface. Its size can changecontinuously, depending on theelectron’s initial condition.One can apply a Bohr-Sommerfeld quantization rule to get quantized orbits, which have dis-crete energies (the Landau levels). The EOM in momentum space, Eq. (69), follows from theeffective Lagrangian,
L(kc; kc) =~
2
2eBkc × kc · B + ~kc · Rc − E(kc). (70)
This gives the generalized momentum,
π =∂L
∂kc
= − ~2
2eBkc × B + ~Rc. (71)
X5.22 Ming-Che Chang
BE(k)
… …
E(k) BEn
Cyclotron orbits Cyclotron orbits
kk
Fig. 10: The quantized cyclotron orbits on two different energy surfaces. The one on the left isa paraboloid near its band edge; the one on the right is a conical surface. Without Berry phasecorrection, the Landau-level energies areEn = (n + 1/2)~ωc andEn = vF
√
2eB~(n+ 1/2)respectively. In graphene, an orbit circling the Dirac point acquires a Berry phase ofπ, whichcancels the 1/2 in the square root.
The quantization condition is given by∮
π ·dkc = (m+γ)h, wherem is a non-negative integerandγ = 1/2 for the cyclotron motion. Therefore, we have
B
2·∮
Cm
(kc × dkc) = 2π
(
m+1
2− Γ(Cm)
2π
)
eB
~, (72)
whereΓ(Cm) =∮
CmRc · dkc is the Berry phase for orbitCm.
This equation determines the allowed size (and therefore energy) of the cyclotron orbit. TheBerry phase correction slightly shifts the Landau-level energies. For example, the orbit aroundthe Dirac point of graphene picks up a Berry phase ofπ due to the monopole at the origin.This cancels the other1/2 in Eq. (72) and results in a zero-energy level at the Dirac point (seeFig. 10). This agrees nicely with experimental measurements [32].
5.2 Non-Abelian generalization
In the one-band theory without internal degrees of freedom,the Bloch state has only one com-ponent and the gauge structure of the Berry phase is Abelian.When the band has internaldegrees of freedom (henceforth simply called the spin), theBloch state has several componentsand the gauge structure becomes non-Abelian. This happens,for example, in energy bands withKramer’s degeneracy. By extending the semiclassical dynamics to such cases, one is able toinvestigate problems involving spin dynamics and spin transport.The scheme for building such a theory is the same as the one in the previous subsection. There-fore, we only give a very brief outline below. One first constructs a wave packet from the Blochstatesψnq,
|W 〉 =
D∑
n=1
∫
BZ
d3qa(q, t)ηn(q, t)|ψnq〉. (73)
Heren is a spinor index for an isolated band withD-fold degeneracy,η = (η1, · · · , ηD)T is anormalized spinor at eachq, anda(q, t) is again a narrow distribution centered atqc(t).Similar to the non-degenerate case, there are three basic quantities in such a formalism, theBloch energyH0(q), the Berry connectionR(q) (and related curvature, now written asF(q)),
X5.23
and the spinning angular momentumL(q) [33]. They all become matrix-valued functions andare denoted by calligraphic fonts. The Bloch energy is simply an identity matrix multiplied byE0(q) since all spinor states have the same energy.The matrix elements of the Berry connection are,
Rmn(q) = i
⟨
umq|∂unq
∂q
⟩
. (74)
The Berry curvature is given by,
F(q) = ∇q × R − iR × R. (75)
Recall that the Berry connection and Berry curvature in the Abelian case are analogous to thevector potential and the magnetic field in electromagnetism(see Sec. 2.1). Here,R andF
also are analogous to the gauge potential and gauge field in the non-AbelianSU(2) gauge fieldtheory [34].The expectation value of the third basic quantity, the spinning angular momentum, is againgiven byL(qc) = 〈W |(r− rc)×p|W 〉. However, it is often written in an alternative (Rammal-Wilkinson) form easier for evaluation,
L(q) = im
~
⟨
∂u
∂q
∣
∣
∣×[
H0 −E0(q)]∣
∣
∣
∂u
∂q
⟩
, (76)
where the cell-periodic function without a subscript is defined as|u〉 =∑D
n=1 ηn|un〉 andH0 isthe Hamiltonian for|u〉. The corresponding matrix-valued functionL therefore has the matrixelements,
Lnl(q) = im
~
⟨
∂un
∂q
∣
∣
∣×[
H0 − E0(q)]∣
∣
∣
∂ul
∂q
⟩
. (77)
Obviously, after taking the spinor average, one has the angular momentum in Eq. (76),L =〈L〉 ≡ η
†Lη =∑
nl η∗nLnlηl.
Equations of motionSo far we have laid out the necessary ingredients in the non-Abelian wave packet theory. Similarto Sec. 5.2, we can use Eq. (59) to get the effective Lagrangian for the center of mass,(rc,kc),and the spinorη. Afterwards, the Euler-Lagrange equation for this effective Lagrangian leadsto the following EOM [33],
~kc = −eE − erc × B, (78)
~rc =
⟨[ DDkc
,H]⟩
− ~kc × F, (79)
i~η =( e
2mL · B − ~kc · R
)
η, (80)
whereF = 〈F〉, and the covariant derivativeD/Dkc ≡ ∂/∂kc − iR. The semiclassicalHamiltonian inside the commutator in Eq. (79) is
H(rc,kc) = H0(kc) − eφ(rc) +e
2mL(kc) · B, (81)
wherekc = qc + (e/~)A(rc).
X5.24 Ming-Che Chang
Even though these equations look a little complicated, the physics is very similar to that of thesimpler Abelian case in Sec. 5.1. There are two differences,however. First, the anomalousvelocity in Eq. (79) is now spin-dependent in general. In some interesting cases (see below),F
is proportional to the spin vectorS = 〈S〉, whereS is the spin matrix. Therefore, if one appliesan electric field to such a system, the spin-up and spin-down electrons will move to oppositetransverse directions. This is the cause of the AHE and the spin Hall effect.Second, we now have an additional equation (Eq. (80)) governing the spinor dynamics. FromEq. (80) we can derive the equation forS,
i~S =⟨[
S,H− ~kc · R]⟩
. (82)
The spin dynamics in Eq. (82) is influenced by the Zeeman energy in H, as it should be. Wewill demonstrate below that the term with the Berry connection is in fact the spin-orbit energy.Such an energy is not explicit inH, but only reveals itself afterH is being re-quantized.
Re-quantizationAs we have shown in Sec. 5.1, re-quantization of the semiclassical theory is necessary whenone is interested in, for example, the quantized cyclotron orbits that correspond to the Landaulevels. Here we introduce the method of canonical quantization, which is more appropriate forthe non-Abelian case compared to the Bohr-Sommerfeld method.In this approach, one needs to find variables with canonical Poisson brackets,
rα, rβ = 0,
pα, pβ = 0,
rα, pβ = δαβ, (83)
then promote these brackets to quantum commutators. As a result, the variables become non-commutating operators and the classical theory is quantized.An easier way to judge if the variables are canonical is by checking if they satisfy the canonicalEOM,
r =∂E
∂p; p = −∂E
∂r. (84)
The variablesrc andkc that depict the trajectory of the wave packet are not canonical variablesbecause their EOM are not of this form. This is due to the vector potential and the Berryconnection,A(rc) andR(kc), in the Lagrangian (see Eq. (63)).In fact, if one can remove these two gauge potentials from theLagrangian by a change ofvariables,
L = p · r −E(r,p), (85)
then these new variables will automatically be canonical. Such a transformation is in generalnon-linear and cannot be implemented easily. However, if one only requires an accuracy tolinear order of the electromagnetic fields (consistent withthe limit of our semiclassical theory),then the new variables can indeed be found.The canonical variablesr andp accurate to linear order in the fields are related to the center-of-mass variables as follows [35],
rc = r + R(π) + G(π),
~kc = p + eA(r) + eB × R(π), (86)
X5.25
whereπ = p+eA(r), andGα(π) ≡ (e/~)(R×B) ·∂R/∂πα. The last terms in both equationscan be neglected in some cases. For example, they will not change the force and the velocity inEqs. (78) and (79). These relations constitute a generalization of the Peierls substitution.When expressed in the new variables, the semiclassical Hamiltonian in Eq. (81) becomes,
H(r,p) = H0(π) − eφ(r) + eE · R(π)
+ B ·[
e
2mL(π) + eR × ∂H0
∂π
]
, (87)
where we have used the Taylor expansion and neglected terms nonlinear in the fields. Finally,one promotes the canonical variables to quantum conjugate variables to convertH to an effec-tive quantum Hamiltonian.The dipole-energy termeE ·R is originates from the shift between the charge centerrc and thecanonical variabler. We will show below that for a semiconductor electron, the dipole term isin fact the spin-orbit coupling.The correction to the Zeeman energy is also related to the Berry connection. Near a band edge,where the effective mass approximation is applicable andE0 = π2/2m∗, this term can be writ-ten aseR · v × B, wherev = π/m∗. We know that an electron moving in a static magneticfield feels an effective electric fieldEeff = v ×B. Therefore, this term arises as a result of theelectric dipole energy in electron’s own reference frame.
Semiconductor electronA necessary requirement for the non-Abelian property is that the Bloch electron has to haveinternal degrees of freedom. In a semiconductor with both space-inversion and time-reversalsymmetries, every Bloch state is two-fold degenerate due toKramer’s degeneracy. But wheredo we expect to see the non-Abelian Berry connection and curvature?Instead of the full band structure, one can start from a simpler band structure using thek · pexpansion. Assuming the fundamental gap is located atk = 0, then for smallk, one has aneffective Hamiltonian with 4 bands, 6 bands, 8 bands, or more, depending on the truncation.In the following discussion, we use a 8-band Kane Hamiltonian that includes the conductionband, the HH-LH bands, and the spin-orbit (SO) split-off band, each with 2-fold degeneracy(see Fig. 11). The explicit Kane Hamiltonian can be found in Ref. [36].We focus only on the wave packet in the conduction band. Without going into details, we firstshow the Berry connection that is essential to the wave packet formulation. The result correctto orderk1 and up to a gauge rotation is [35],
R =V 2
3
[
1
E2g
− 1
(Eg + ∆)2
]
σ × k, (88)
whereV = ~
m〈S|px|X〉, Eg is the energy gap, and∆ is the SO gap. Therefore, the dipole term
eE · R in Eq. (87) becomes,
Hso = eE · R = αE · σ × k, (89)
whereα ≡ (eV 2/3)[1/E2g −1/(Eg +∆)2]. It coincides precisely with the spin-orbit coupling of
a conduction electron. This shows that the SO coupling has a very interesting connection withthe Berry connection. This is also the case for the SO coupling in Dirac’s relativistic electrontheory [37].
X5.26 Ming-Che Chang
4-band Luttingermodel
8-band Kane model
E(k)
Eg
∆HH
LH
SO
CB
Fig. 11: One can use the 4-band Luttinger model or the 8-band Kane model to approximate theenergy bands near the fundamental gap.
The Berry curvature calculated from Eq. (75) gives (to the lowest order)F = α/eσ, which isproportional to spin. Therefore, the anomalous velocityeE × F in Eq. (79) isαE × 〈σ〉. Thatis, spin-up and spin-down electrons acquire opposite transverse velocities. In non-magneticmaterials, these two species have the same population and wedo not expect to see a net trans-verse current. However, “if” one defines a spin current as thedifferenceof these two transversecurrents, then there will be a netspincurrent, giving rise to the spin Hall effect [38].One can also calculate the spinning angular momentum of the conduction electron from Eq. (77).The result is,
L = −2mV 2
3~
(
1
Eg− 1
Eg + ∆
)
σ. (90)
Through the Zeeman energy in Eq. (87), the orbital magnetic moment generated from Eq. (90)contributes an extrag-factor,
δg = −4
3
mV 2
~2
(
1
Eg
− 1
Eg + ∆
)
. (91)
This is the anomalousg-factor of the conduction electron [39]. Therefore, the anomalousg-factor in solid is indeed a result of the self-rotating motion of the electron wave packet.Finally, the effective quantum Hamiltonian in Eq. (87) for the conduction band has the followingform,
H(r,p) = E0(π) − eφ(r) + αE · σ × π +δg
2µBB · σ, (92)
whereE0 includes the Zeeman energy from the bare spin,α is given below Eq. (89),δg is givenin Eq. (91), and the correction to the Zeeman energy has been neglected. This Hamiltonianagrees with the one obtained from block diagonalization [36]. The wave packet approach is notonly simpler, but also reveals the deep connections betweenvarious effective couplings and theBerry potential.Some comments are in order: First, we emphasize again that itis necessary to include theBerry curvature and orbital moment in order to account for physical effects to first order in
X5.27
external fields. Furthermore, from the discussions above, we can see that these quantities arealso sufficient for building a correct quantum theory.Second, starting from a quantum theory, one can construct a semiclassical theory in a specificsubspace. This theory can later be re-quantized. The re-quantized effective theory applies to asmaller Hilbert space compared to the original quantum theory. Nevertheless, it can still haveits own semiclassical theory, which in turn can again be re-quantized. As a result, a hierarchyof effective theories and gauge structures can be produced,all within the wave packet approach(see Ref. [35] for more discussions).
6 Concluding remarks
In this review, selected topics related to Berry phase in solid state physics are reported. Manyof these topics have been fully developed over the years. Theexposition here only serves asan introduction, without going into details and more recentdevelopment. Readers interestedin certain topics can consult some of the following books or review articles: [1] and [40] onBerry phase in general, [41] and [42] on electric polarization, [43] on quantum Hall effect, [44]and [45] on anomalous Hall effect, [46] and [47] on dynamics of Bloch electrons, and [35] onnon-Abelian wave packet dynamics.In optics, the Berry curvature is related to a transverse shift (side jump) of a light beam reflectedoff an interface.[48] The shift is roughly the order of the wavelength. Its direction dependson the circular polarization of the incident beam. This is called theoptical Hall effect, or theImbert-Federov effect,[49] which is not covered here. The side jump of a light beam is similar tothe analogous “jump” of an electron scattering off an impurity in the anomalous Hall effect [22].A more detailed study of the optical transport involving spin can be found in Ref. [50].Several topics not covered here can be found in an upcoming review on Berry phase in solid statephysics [51]. These topics include the orbital magnetization of a solid, dipole moment of thewave packet, anomalous thermoelectric transport, and inhomogeneous electric polarization. It isamazing that the Berry phase plays such a versatile role in somany solid-state phenomena. Onthe other hand, several challenging subjects still remain largely unexplored. For example, theeffect of the Berry phase in systems in which non-adiabatic processes or many-body interactionis crucial. Therefore, one can expect to see more of the intriguing Berry phase effects in solidstate systems.
X5.28 Ming-Che Chang
References
[1] A. Shapere and F. Wilczek, (ed)Geometric Phases in Physics(Singapore: World Scien-tific, 1989)
[2] For example, see p. 290 in L. I. Schiff,Quantum Mechanics(McGraw Hill, 3rd ed 1968)
[3] M. V. Berry, Proc. R. Soc. A392, 45 (1984)
[4] B. F. Schutz,Geometrical methods of mathematical physics(Cambridge University Press,1980)
[5] M. Stone, Phys. Rev. D33, 1191 (1986)
[6] For example, see K. Ohgushi, S. Murakami, N. Nagaosa, Phys. Rev. B62, R6065 (2000);Y. Taguchi, Y. Oohara, H. Yoshizawa,et al. Science291, 2573 (2001); R. Shindou, N.Nagaosa, Phys. Rev. Lett.87, 116801 (2001)
[7] L. P. Levy et al. , Phys. Rev. Lett.64, 2074 (1990); V. Chandrasekharet al. , Phys. Rev.Lett. 67, 3578 (1991)
[8] D. Loss, P. Golbart, and A. V. Balatsky, Phys. Rev. Lett.65, 1655 (1990)
[9] F. Schutz, M. Kollar, and P. Kopietz, Phys. Rev. Lett.91, 017205 (2003)
[10] D. Loss, D. P. DiVincenzo, and G. Grinstein, Phys. Rev. Lett. 69, 3232 (1992); J. vonDelft, and C. L. Henley, Phys. Rev. Lett.69, 3236 (1992)
[11] See Chap 21 in R. Shankar,Quantum Mechanics(Springer, 2nd ed. 1994)
[12] Y. Zhanget al. , Nature438, 201 (2005).
[13] R. Resta, Ferroelectrics136, 51 (1992)
[14] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B47, 1651 (1993)
[15] D. J. Thouless, Phys. Rev. B27, 6083 (1983); Q. Niu, Phys. Rev. Lett.64, 1812 (1990)
[16] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980)
[17] R. B. Laughlin, Phys. Rev. B.23, 5632 (1981)
[18] These mobile charges are carried through an extended state in the middle of every Landaulevel. Such an extended state is flanked by localized states on both sides of the energy.When the chemical potential falls within the localized states for a range of magnetic field,no charges can be transported and the Hall conductance exhibits a plateau over this rangeof magnetic field. Without disorder, there will be no localized states, thus no plateaus.Therefore, disorder also plays a crucial role in the quantumHall effect.
[19] J. D. Thouless, M. Kohmoto, P. Nightingale, and M. den Nijs, Phys. Rev. Lett.49, 405(1982); Also see M. Kohmoto Ann. Phys. (N.Y.)160, 355 (1985)
X5.29
[20] Because the vector potentialA0 in the unperturbed HamiltonianH0 breaks the latticesymmetry, one needs to solve the Schrodinger equation based on the so-called magnetictranslation symmetry. Therefore, to be precise, the Bloch state, the Bloch energy, and theBrillouin zone mentioned here for the QHE should actually bethe magnetic Bloch state,the magnetic Bloch energy, and the magnetic Brillouin zone,respectively. This distinctionis not emphasized in the text.
[21] Q. Niu, D. J. Thouless, and Y. S. Wu, Phys. Rev. B31, 3372 (1985)
[22] L. Berger and G. Bergmann, inThe Hall Effect and Its Applicationsedited by C. L. Chien,and C. R. Westgate (Plenum, New York, 1979) p. 55.
[23] Another often mentioned explanation is the side-jump mechanism proposed by Berger in1970. It involves the impurity scattering but the underlying formalism can be rephrased ina way related to the Berry connection. It also predictsρH ∝ ρ2
L. One can consult Ref. [22]for more details.
[24] R. Karplus and J. M. Luttinger, Phys. Rev.95, 1154 (1954)
[25] J. Smit, Physica21, 877 (1955)
[26] Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science291, 2573(2001); T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev.Lett. 88, 207208 (2002)
[27] Z. Fanget al. , Science302, 92 (2003)
[28] C. Zeng, Y. Yao, Q. Niu, and H. Weitering, Phys. Rev. Lett. 96, 37204 (2006)
[29] M. C. Chang, and Q. Niu, Phys. Rev. B53, 7010 (1996); G. Sundaram and Q. Niu, Phys.Rev. B59, 14915 (1999)
[30] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett.97, 026603 (2006).
[31] P. C. Chuu, M. Chang, and Q. Niu to be published
[32] S. E. Novoselovet al. , Nature438, 197 (2005); Y. Zhanget al. , Nature438, 201 (2005)
[33] D. Culcer, Y. Yao, and Q. Niu, Phys. Rev. B72, 85110 (2005)
[34] F. Wilczek, and A. Zee, Phys. Rev. Lett.52, 2111 (1984)
[35] M. C. Chang, and Q. Niu, J. Phys.: Condens. Matter20, 193202 (2008)
[36] R. Winkler Spin-orbit coupling effects in two-dimensional electron and hole systems(Springer, 2003)
[37] H. Mathur, Phys. Rev. Lett.67, 3325 (1991); R. Shankar, and Mathur, Phys. Rev. Lett.73,1565 (1994)
[38] S. Murakami, N. Nagaosa, and S. C. Zhang, Science301, 1348 (2003)
[39] See Problem 9.16 in P. Y. Yu and M. CardonaFundamentals of Semiconductorsthe thirded. (Springer, 2003)
X5.30 Ming-Che Chang
[40] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger,The Geometric Phasein Quantum Systems(Springer-Verlag, Heidelberg, 2003)
[41] R. Resta, Rev. Mod. Phys.66, 899 (1994)
[42] R. Resta, J. Phys.: Condens. Matter12, R107 (2000)
[43] R. E. Prange, and S. M. Girvin,The quantum Hall effect(Springer, 2nd ed., 1990)
[44] N. Nagaosa, J. Phys. Soc. Jpn.75, 042001 (2006)
[45] N. A. Sinitsyn, J. Phys: Condens. Matter20, 023201 (2008)
[46] E. I. Blount,Solid State Physics, edited by Seitz F and Turnbull D (Academic Press Inc.,New York, 1962), Vol. 13
[47] G. Nenciu, Rev. Mod. Phys.63, 91 (1991)
[48] M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett.93, 83901 (2004); K. Sawada,and N. Nagaosa, Phys. Rev. Lett.95, 237402 (2005); M. Onoda, S. Murakami, and N.Nagaosa, Phys. Rev. E74, 66610 (2006)
[49] F. I. Fedorov, Dokl. Akad. Nauk SSSR105, 465 (1955); C. Imbert, Phys. Rev. D5, 787(1972);
[50] K. Y. Bliokh et al., Phys. Rev. Lett.96, 073903 (2006); K. Bliokh, Phys. Rev. Lett.97043901 (2006); C. Duval et al., Phys. Rev. D74, 021701 (R) (2006)
[51] D. Xiao, M. C. Chang, and Q. Niu, to be published
